Question

Consider the following two assets. Returns on asset l has a mean of μ i and standard deviation of . Returns on asset 2 has a mean of μ2 and standard deviation of σ2. The correlation coefficient 2 measures how the two assets' returns are correlated, and it takes on values between-1 and +1. An investor puts Wi fraction of her wealth into stock 1, and W2 = 1-WI fraction of her wealth into stock 2. 1. Using the equation on the standard deviation of portfolio returns, discussed in today's class, argue that the portfolio risk is increasing in P12- 2, when ρ1,2 1, argue using mathematical frmulas that the portfolio standard deviation is equal to the weighted average of the standard deviation of the individual stocks in the portfolio. (Hnt, Square of Summation) 3. Now suppose ρ1,2- 1 . If the investor wants to minimize her risk in investing in this portfolio, how should she choose Wi and W2-1-W? (Hint, Sqare of Difference

Answer 1

Portfolio variance=w1^2*sd1^2+w2^2*sd2^2+2*w1*w2*correl(1,2)*sd1*sd2
Portfolio standard deviaiton=sqrt(variance)
partially differentiaitong variance w.r.t. correl(1,2)
=2*w1*w2*sd1*sd2
So, when w1 and w2 are more than zero, the above calculated slope be positive implying increase in variance when correl(1,2) increases and when variance increases standard deviaition would also increase

When correl(1,2)=1
Portfolio standard deviaiton=sqrt(w1^2*sd1^2+w2^2*sd2^2+2*w1*w2*sd1*sd2)=w1sd1+w2sd2, which is the weighted average of standard deviaition of individual stocks in the portfolio

When correl(1,2)=-1
Portfolio standard deviaiton=sqrt(w1^2*sd1^2+w2^2*sd2^2-2*w1*w2*sd1*sd2)=w1sd1-w2sd2
To minimize risk, w1sd1-w2sd2=0
As w2=1-w1
=>w1/(1-w1)=sd2/sd1
=>(1-w1)/w1=sd1/sd2
=>1/w1-1=sd1/sd2
=>1/w1=(sd1+sd2)/sd2
=>w1=sd2/(sd1+sd2)